Practice Test 4

Problem (1) Using Mathematical Induction prove,  

Problem (2) Using Mathematical Induction prove,  

Problem (3) Show that the product of two integers of the form is again of this form.

Problem (4) Show that the product of two integers of the form is of the form .

Problem (5) Prove that each integer of the form has a prime factor of this form.

Problem (6) Prove that there are infinitely many primes.

Problem (7)  Show that every positive integer greater than 6 is the sum of two relatively prime integers greater than 1.

Problem (8) Show that for any integer one of is divisible by

Problem (9) Apply the Euclidean Algorithm to find

Problem (10) Apply the Euclidean Algorithm to find

Problem (11) Find the unique factorization of

Problem (12) Find the unique factorization of

Problem (13) Determine how many solutions to

Problem (14) Find all solutions to

Problem (15) Use mathematical induction to show that the sum of the cubes of three consecutive integers is divisible by 9.

Problem (16) Apply the Euclidean Algorithm to solve Then find all solutions.

Problem (17) Show that if and are integers such that and then

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